We define an (uncurved) \(A_\infty \)  functor to be a quasi-equivalence if it induces an equivalence on the level of cohomology, and say that two \(A_\infty \)  categories are quasi-equivalent if they are connected by a zig-zag of quasi-equivalences (Definition REF ).
This is a `good' equivalence relation if the morphism cochain complexes are cofibrant (see [1]}); this is true for example if the coefficient ring \(\)  is a field, but we have not checked it in general.
